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THz Emitters and Detectors

1. 1 0357554 0 7816893 0 018003440 0 016067891 1 2479989 0 8403893 9 9 indices 0 00 0 50 1 00 1 50 2 00 2 50 3 00 wavelength microns Figure A 1 Refractive indices for BBO Page 61 of 81 B Visual Basic functions for phase matching calculations The Visual Basic functions for Type II phase matching used in Excel are listed here Functions for Type I downconversion were also written though these are fairly trivial and are not listed For clarity of reading comments are type set in italic and Visual Basic keywords in bold In order to use these functions the inverse trigonometric functions arccos and arcsin were also be defined as only arctan is defined in the version of Visual Basic used Sellmeier indices Const A e As Double 1 5920433 extraordinary Const B e As Double 0 7816893 Const C e As Double 0 01600607891 Const D e As Double 0 8403893 Const E e As Double 91 Const A o As Double 1 7018379 rordinary Const B o As Double 1 0357554 Const C o As Double 0 01800344 Const D o As Double 1 2479989 Const E o As Double 91 Speed of light microns per sec Const C As Double 299000000000000 Const pi As Double 3 14159265358979 Public Function NE L As Double As Double calculates the extraordinary refractive index of BBO as a function of wavelength i e half the extraordinary axis of the index ellipsoid Arguments i L wavelength in microns NE S Sqr A se Bee y Ch
2. black body source Figure 4 2 Schematic of spectral radiance measurement set up using Type II collinear downconversion The CuBB operates by melting very pure copper surrounding the cavity itself see Figure 4 3 and then cooling it below its freezing temperature Due to the purity of the copper it will under cool without freezing as there are few seed sites to initialise the Page 47 of 81 crystallisation As it does freeze the temperature rises back up to the freezing point temperature 1 358 x 10 K where it remains stable for some time By Wien s Law the maximum emission at this temperature is at 2 134 um External furnace cavity heater coils r aperture either graphite graphite metal g copper or rhodium 99 9999 pure Figure 4 3 Generalised schematic of a fixed point blackbody The blackbody radiation is emitted from the central cavity held at a constant temperature by the freezing of the very pure metal surrounding it Based on a sketch by E Woolliams In order to image the blackbody cavity and collimate the beam we propose to use an off axis parabolic mirror OAPM This is a section taken from a parent parabolic mirror that enables light from a point source to be collected collimated and turned by a fixed angle Figure 4 4 shows a schematic of an OAPM indicating the meaning of the terms real and apparent focal length Page 48 of 81 real focal length turning apertures angle
3. o 100 10 a 1 60 70 80 90 100 110 120 130 140 time ns Averaged counts delay 104 ns 100000 10000 1000 3 s 2 c O o 100 b 60 70 80 90 100 110 120 130 140 time ns Figure 3 6 Coincidence counts for the two measurement runs with delays of a 72 ns and b 104 ns In both cases the number of triggers is 10 000 000 The data is averaged over several runs and binned The higher noise in a is due to averaging fewer runs Note the reflection at around 121 ns in a Also note that the ordinate scales are logarithmic Page 40 of 81 3 6 1 Values and uncertainties of photon counts Nc is calculated by integrating over the coincidence peak typically over 15 ns either side of the centre We then integrate a region away from the peak and any reflection peaks over the same time span to find the accidental counts To find the false counts we first measure the counts per second with the PDC on to calculate how long it would take to generate 10 000 000 triggers With the PDC off the false counts per second are found and then multiplied by this time period As we are assuming that photon arrivals are governed by Poisson statistics the uncertainty in the false counts i e the standard deviation is multiplied by the square root of this period Table 3 2 and Table 3 3 show results for two separate sets of measurements the figures for the quantum efficiency are intermediate results without the transmittance of the DUT c
4. outlined above to minimise the error we might assume initially that this uncertainty 1s negligible Ultimately it will be the reproducibility and repeatability of the measurements that will determine whether this 1s so 3 2 3 Other uncertainties Not included in 3 5 are any uncertainties due to systematic errors that might be introduced by the particular experimental set up or contingent software and electronics In our experiments we found that impedance mismatching on the coincidence counters created reflections in the BNC cables giving rise to additional counts This is not a particular problem so long as the reflections do not occur so Page 31 of 81 rapidly as to overlap with the coincidence peak However their presence must still be taken into account when summing the coincidence counts as they may produce a much greater background reading The technical aspects of this problem are discussed in the next section 3 3 Experimental set up The experimental set up for the quantum efficiency measurements 1s shown in Figure 3 2 An argon ion laser provides vertically polarised pump photons at 351 1 nm In our set up the beam is passed through a narrow band pass UV filter 351 1 nm which excludes any plasma radiation from the laser into an enclosed box providing a light tight environment for the experimental apparatus and reflected by three mirrors to bring it on line with the optical rail resulting in a beam with horizontal po
5. x Note that as di5 d3 when Kleinman Symmetry is valid some authors for instance xiv write the above expression with d3 in place of di5 A further notational complication arises since there is not universal agreement about the orientation of the crystal x y axes Some authors have these rotated by 90 relative to those used above This requires a transformation of the d tensor and 2 27 becomes xv d 8in cos cos3 Type I alternative 2 28 ff where d _ a3 dis di1 d2 and p Dp 90 For Type I downconversion we take k to be the extraordinary wave The unit vectors are NOW e osO cosge osA singe md e ng e ospe 2 29 e os cos e os sing e n0 which gives d iz os O sin sin g cos sin sing 5 eff T N C0S cos cos g nt Type IT 2 30 Page 24 of 81 we get the same result for both downconverted waves although this 1s not obvious from the maths If we once more take the wave vectors and optic axis to be coplanar we arrive at the result often given in nonlinear optics literature dg l cos 8 cos3 Type II 2 31 Again alternative notation exists To solve equations 2 26 and 2 30 we first need to solve the phase matching equations 2 19 and 2 20 for a and a2 We can then transform from the pump frame of reference Figure 2 4 to the crystal frame Figure 2 7 for each wave to find the required
6. Figure 2 2 a Type II down conversion for a negative uniaxial crystal The optic axis lies in the plane of the paper b The collinear case where at the point where the two light cones touch the downconverted photons are collinear with the pump photon In order to calculate the phase matching conditions we note the relationship between the magnitude of the k vector and the angular frequency epee eee 2 17 where Ois the angle between the k vector and the optic axis For waves polarised along the ordinary axis the refractive index no does not depend on Figure 2 3 illustrates the dependence of the extraordinary refractive index n at a given wavelength Here n and ne are the semi major and semi minor axes respectively of Page 16 of 81 the projection of the index ellipsoid on to the plane including the optic axis The wavelength dependence of these refractive indices can be found from appropriate Sellmeier equations see Appendix A Figure 2 3 The index ellipsoid for a negative uniaxial crystal viewed side on from the negative y direction showing how the extraordinary index of refraction n varies with the angle of propagation direction k to the optic axis z axis n is the semi major axis and n is the semi minor axis Using Figure 2 3 we can find an expression for ne 6 1 2 pio now lto __ 2 18 g g Ln _ n o tan Finally by expressing the k vectors of the signal and idler wi
7. OAPM eee apparent focal length Figure 4 4 Schematic of an off axis parabolic mirror The shaded region represents the mirror sectioned from the parent the parabolic curve shown Since the distance from the outside aperture to a point in the cavity in the CuBB is about 400 mm the apparent focal length needs to be greater than this The size of the CuBB also means that in order to make room for it in the laboratory it would be practical to have it lying alongside the optical bench and use a 90 turn on the OAPM Unfortunately off the shelf OAPMs with this apparent focal length and turning angle are not available This 1s because the parent parabolic mirror would have to be exceptionally large Instead such a mirror would have to be fabricated as a stand alone mirror Since high quality off the shelf mirrors cost thousands of dollars this is likely to be a very expensive option Another consideration must be the coating on the mirror For short wavelengths aluminium may be sufficient but longer than 2 microns we would have to use gold Page 49 of 81 coatings The choice of wavelength may therefore affect our mirror specifications Naturally we would also need to characterise the transfer function of the mirror 4 3 Downconversion options For the spectral radiance measurements we had a second BBO crystal available cut with the optic axis at 49 2 to the normal and rotated by an azimuthal angle of 30 as marked
8. spectral radiance measurement we would want to measure an arbitrary source Therefore we need to calculate the transfer function of the imaging optics so that we Page 53 of 81 can extrapolate back from any results to the original source and ascertain how accurately this can be done 4 4 4 Overlap factor and transmission losses As in the QE measurements we also need to take the transmission losses in the crystal and any other optics into account Depending on the experimental set up analysing the transmission of downconverted light may be more complicated than for the QE measurements In the latter case we assumed that downconversion takes place equally throughout the thickness of the crystal so we could take the centre as the average point This remains a valid assumption for spontaneous PDC but for stimulated PDC it will depend on how well the signal overlaps the pump and the downconverted modes This overlap factor will involve three components Firstly how well the source overlaps spatially with the pump This will determine the rate of any downconversion Secondly the signal must overlap spectrally with the downconverted light in order to achieve parametric amplification at the required frequencies Thirdly the source must overlap angularly with the downconverted light These last two points are of course related and correspond to the phase matching requirements The first of these components has been calculated by Migdall
9. supply 5V Low level sink current gt 90mA a DN a O aa A Page 77 of 81 Detector pulse shapes before attenuation and inversion Detector pulse waveforms SPCM AQR 14 SPCM AQR 13 signal V time ns Figure C 1 The waveforms of the APD detectors used These graphs were obtained using an oscilloscope Page 78 of 81 8 11 iii iv LV vi vii viii References Aspect A Grangier P Roger G Experimental Tests of Realistic Local Theories via Bell s Theorem Physical Review Letters 47 460 1981 Fox N P Primary radiometric quantities and units Metrologia 37 507 513 2000 Klyshko D N Utilization Of Vacuum Fluctuations as an Optical Brightness Standard Sov J Quant Electron 7 591 595 1977 Rarity J G Ridley K D Tapster P R Absolute Measurement of Detector Quantum Efficiency Using Parametric Downconversion Appl Opt 26 4616 4619 1987 Migdall A L Datla R U Sergienko A Orszak J S Shih Y H Absolute Detector Quantum Efficiency Measurements Using Correlated Photons Metrologia 32 479 483 1995 96 Migdall A Datla R Sergienko A Orszak J S Shih Y H Measuring Absolute Spectral Radiance with Correlated Visible Photons Technique Verification and Measurement Uncertainty Appl Opt 37 3455 3463 1998 Brida G Castelletto S Degiovanni I P Novero C Rastello M L Quantum Efficiency and D
10. the detector further from the crystal In our set up the detectors are both mounted on automated stages which facilitates systematic fine tuning of the alignment by maximising the coincidence counts The alignment of the DUT with respect to the lens can also be adjusted remotely to improve the focussing We must also contend Page 30 of 81 with the fact that the PDC does not arise from a point source but takes place over an extended area The use of thinner crystals can alleviate this Since the wavelengths of the downconverted photons define the half cone angle with respect to the pump beam the use of filters may seem superfluous However spectral selectivity is useful in reducing false triggers which are due to the fluorescence at the crystal and the spread of downconverted light around the wavelength of interest see Figure 2 5 Using a filter over the DUT may not be necessary if we can subtract the accidental counts accurately from the coincidences This is admissible providing the contribution to the uncertainties due to the standard deviation of the accidental counts is negligible If a filter 1s used we need to ensure that its transmission bandwidth encompasses that of the trigger filter Figure 3 7 shows the transmittance curves of the filters used in these experiments Quantifying the uncertainty due to geometric misalignment is somewhat difficult Having made our experimental apparatus as robust as possible and taken the steps
11. the crystal this approximation is accurate to 0 01 or better if the reflectances at both faces are similar 3 2 2 2 Losses in optical elements Filters were used to pass the downconverted light whilst blocking any stray light due primarily to fluorescence Lenses were also required to focus the light on to the detector active areas At present laser techniques have achieved accuracies of 0 01 to 0 02 for the transmittance measurements of spectral filters at NPL and 0 02 for lenses xviii Page 29 of 81 Table 3 1 summarises the currently achievable measurement uncertainties in the visible region The figures are taken from xvii and are based on work undertaken at NPL The figures suggest that overall uncertainties due to optical losses can be as low as 0 05 Table 3 1 Measurement uncertainties currently achievable taken from ref xvii Measurement uncertainties O ee ae poe ee o A 3 2 2 3 Geometric and spectral misalignment We need to ensure that the twins of all correlated photons at the wavelength of interest incident on the trigger are also incident on the DUT Reasons why this may fail to be the case are geometric misalignment poor focussing on to the detector active areas or poor spectral overlap on the filters used In practice we can improve the overlap between the detectors by making the trigger subtend a smaller angle from the PDC source either by use of a narrower aperture or by physically placing
12. the order 0 38 is found which appears to be in line with the specifications in Appendix E Page 44 of 81 4 Spectral radiance 4 1 Spectral radiance in terms of fundamental constants We saw in section 2 2 that the quantum mechanical analysis of PDC predicted spontaneous downconversion which was interpreted as being stimulated by one photon per mode background fluctuations We now consider a source of signal photons incident on a pumped medium such that the phase matching conditions are met We will now have parametric amplification of the signal together with a correlated growth in the idler fields see Figure 4 1 Assuming that there is no initial input at the idler frequency then from 2 16 we have with the signal on N Gor G No sinh x 4 1 With the signal off lt Nso gt 0 so taking the ratio of lt N t gt oy and lt N t gt orr and rearranging we have NOn WNG N g 4 2 Although this equation is dimensionless numerically lt N o gt is the average number of photons per mode of the initial signal field Now a spectral radiance of one photon per mode can be written v1 Ro 4 3 Page 45 of 81 Thus by counting the photons in the idler field we can make an absolute measurement of the signal spectral radiance in terms of the fundamental constants h Planck s constant and c the speed of light A spin off from this technique is that where the signal may be in the infrared we can arrange the
13. the stages can be moved to tweak the alignment The Enable stages button must toggled first to do so Note that it is safer to leave this off if the alignment is fine to prevent accidental movement Page 71 of 81 D Photographs of the experimental set up for QE measurements Figure D 1 The light tight experimental enclosure The argon laser can be seen in the background on the left On top of the enclosure is the PC and counting electronics Figure D 2 View from the end of the optical table with the side panels removed looking from the detectors in the boxes in the foreground towards the crystal The beam enters through a narrow band pass filter over an opening in the far panel Page 72 of 81 4 h O a Figure D 3 Detector assembly trigger with the light tight cover removed The filter is fixed to the front plate The lens is mounted on the translation stage in the middle On the right is the detector mounted on x y and z translation stages Figure D 4 The trigger detector enclosed in its light tight box shown mounted on the automated rotation and translation stages Page 73 of 81 Figure D 5 Mountings on the optical rail On the left is the polariser off rail The HWP is somewhat obscured behind the iris mountings Also obscured are the mirrors for reflecting the laser beam along the optical rail The beam enters from the left The crystal mounting is on the right The HeNe laser just seen on the r
14. the trigger corresponding to the set delay An EG amp G Ortec 994 dual counter timer is used to make independent counts of each channel Both counters were read remotely via LabView modules incorporated into the software The 9308 required NIM inputs whilst the detectors produced TTL pulses A TTL NIM converter unit was available but it was not performing properly so instead a NIM pulse was simulated by attenuating the signal to 0 8V measured on an oscilloscope with a 50Q input impedance and inverting it to give a negative leading edge Testing the 9308 with a 0 4V amplitude square wave offset to 0 4V to reproduce this we found that we recorded all the stop events at 100kHz Despite these efforts a problem still remained with impedance mismatching leading to pulse reflections However the reflections were not excessive and at the time of writing a Page 34 of 81 new bespoke TTL NIM converter unit has been commissioned which will hopefully rectify the problem 3 4 Aligning the detectors 3 4 1 Pre alignment of detectors and lenses The detectors were initially aligned by eye positioning them so that they lie on a line at the calculated half cone angle of the PDC from the crystal An integrating sphere was then placed at the position of the crystal and the lens detector assemblies adjusted so that the focussed light was under filling the detector active areas With the laser on the PDC can be checked for by imaging the li
15. then optimise the mirror alignment for stimulated PDC At present the main focus of work for improving these techniques with correlated photons lies in realising better uncertainties for transmission losses in the crystal ensuring good alignment of the experimental apparatus and developing effective Page 59 of 81 procedures for achieving this Using current techniques for transmittance measurements there seems to be no reason in principle why improvements of an order in magnitude might not be achieved In order that these uncertainties should be on a par with those currently realised in cryogenic radiometry further improvements would be needed 6 Acknowledgements The author would like to acknowledge the following for their help and participation in this work Dr Chris Chunnilall project supervisor Dr Jessica Cheung co worker and co author of the software John Mountford for writing the TP stage drivers as well as advice and assistance with the electronic equipment Emma Woolliams for general advice on metrology and Dr Leon Rogers for facilitating the student placement at NPL Page 60 of 81 7 Appendices A Sellmeier equations for BBO The refractive indices for BBO used in this report are calculated using the following Sellmeier equation due to 20 where the Sellmeier constants are given in the table below f B D i A 1 Eu E EA Table A 1 Sellmeier constants for BBO o ray e ray 1 7018379 1 5920433
16. way is Ne ft urliigl pur 3 2 If we know Nrig it is a straightforward task to calculate the quantum efficiency of the DUT Ne Lpur N trig _ur 3 3 In practice Nc will be increased by stray photons and dark counts Similarly the total trigger counts will include extra counts giving rise to false triggers which must be subtracted from N ic Denoting these extra counts by Nace and Nase respectively we can re write 3 3 as No N acc 6 T i DUT trig false 3 4 Page 27 of 81 3 2 Sources of uncertainty In assessing the accuracy of this method the sources of uncertainty need to be addressed The fractional uncertainty in pur is given by 2 AN yr Tan WN AN jaise i V 3 5 acc 1 ur gt 4 ny D L Ve N ae p N es l 3 2 1 False and accidental counts The uncertainty in Nace arises because in practice we have to estimate this quantity and take an average over several readings Similarly as there is fluctuation in the number of coincidence counts over a number of runs Nc must also be arrived at through averaging The number of false counts must be extrapolated from the counts per second with no PDC to the counts expected over the time it takes to register a given number of triggers Note that in our experiment since the number of triggers that we count for is fixed the uncertainty in Nig 18 Zero If we assume that the photon arrival times are Poisson distributed
17. 23 i T E 7 TDK NDE l ESE 5 BGS 26 5 6 2 S We now take the dot product of 2 22 with the unit vector in the direction of polarisation for the frequency component we are interested in with 2 23 consisting of the mixing terms for the other two components Figure 2 7 shows the pump and downconverted wave vectors 1 and 2 in relation to the crystal axes z being the optic axis For extraordinary waves the polarisation will be in the plane defined by the optic axis and the wave vector The polarisation of ordinary waves will be perpendicular to this plane Thus for Type I downconversion the polarisation unit vectors for each field will be e in e OS Pe e ing e os pe 2 24 e os cosge os sing e nd where the magnitudes of each field will be E gt and E Taking the to be the field of interest we find Page 22 of 81 e Pwr d EE 2 25 where the effective nonlinearity is d ef s8In cos Q 22 COSA sin g 4 i Type I 2 26 Clearly since interchange of subscripts 1 and 2 makes no difference we have the same result for either ordinary wave Figure 2 7 The wave vectors for the pump p signal and idler 1 and 2 shown in relation to the crystal axes z is the optic axis In the case where all the waves and the optic axis are coplanar i and we have d oy sin 0 COS 0 sin 30 Type I 2 27 Page 23 of 81 as quoted in
18. Correlated Two Photon Metrology M P Vaughan A project conducted at the National Physical Laboratory Queens Road Teddington TW11 O0LW UK and submitted as a dissertation for the MSc in the Physics of Laser Communications Dept of Electronic Systems Engineering University of Essex 2003 Abstract Research has been ongoing in the field of quantum metrology using correlated photons to establish radiometric scales in the photon counting regime without the need of calibrated detectors or sources Currently detector and source scales are traced ultimately to the SI unit of electricity The use of correlated photons offers methods for establishing these scales absolutely This removes the need for long calibration chains for the determination of detector quantum efficiency QE and enables radiation scales to be based on fundamental constants A further advantage in the determination of spectral radiance 1s the possibility of using detectors in the visible range to measure sources in the IR The focus of current work is to establish and improve the levels of uncertainty associated with these methods in order that they become viable techniques Page 2 of 81 Table of Contents BE O11 Gl ean ee ree eRe OA REET eRe re A ee Ee Ne weve ERI ene Peae Ce Ne ween ve EEN 2 F A OCU CUO aar A 5 Zi Farametric down Conversion ziosire esae R E E 8 2 1 POC as three wave Mie so escs eh catia teeter nastiest EE 9 2 2 Ouantuin mechanical anal ys1S
19. O DAO A O r exp c ae i 2 4 with a similar expression for the idler frequency Equation 2 4 can put into scalar form by expressing the second order polarisation component in a particular direction in terms of an effective nonlinearity dey see section 2 4 We then have PO T 6 d A A exp H Saaie 2 5 where the frequency dependence of the amplitudes is now implied by subscripts to simplify the notation Substituting 2 5 into 2 2 and taking the direction of propagation to be along the z axis gives for both the signal and the idler the coupled equations A D y A exp AN D Z 2 6 0 A D n e yA expt Az where Page 10 of 81 D U gt e TAN i 2 7 D U ea E ia and A a T i 2 8 We are assuming that the pump is not depleted so 4 z is taken as a constant Comparison with the second equation of 1 1 shows that the phase mismatch term Ak is just a restatement of the conservation of momentum Had this analysis been performed in the temporal domain the conservation of energy would also have emerged Solving equations 2 6 with the boundary conditions 4 0 Aso 4 0 Ajo and putting Ak 0 we arrive at a A 4 cosh z S A sinh z _ Ta 2 9 Paa mee A 1 cosh z 1 A sinh z ed where the gain term g 1s given by g y 2 209 IR 2 10 4c n n Page 11 of 81 c is the speed of light and ns n are the refractive indice
20. a spreadsheet file with the trigger counts in the first column and the DUT counts in the second The pulse specs box on the right hand side of the window sets the parameters for the 994 counter VI The counts are divided by the pulse width to give the counts per second Page 67 of 81 Clicking on EXIT closes the VI and takes you back to the main window Scanning over an axis To scan over an axis select the required axis in the Which Stage box and click on the SCAN button This opens the window shown in Figure C 2 Which axis Trigger amp axis Scan type 994 Trigger count ve limit Scanning a J TAD 7 a e limit F a Increment y0 05 STOP E Counts step 1 a Number of triggers ic Fi Offset5ec 6 0000E 8 3 Figure C 2 The Scan Axis window Counts Current pos 16 60 40000 0 LES i E wa aE Dr Ha wa aE pr a ET ERA a a T e o 1iditttiidtt LLII Pi 12 60 14 00 15 00 16 00 16 60 Position Current count error auk 40620 status code F go SOUPCE The scans can either read from the 994 counter either the Trigger or the DUT or the 9308 counter Ensure that these counters are connected to the detectors correctly before continuing The ve and ve limit boxes give the limits of the scan around the position started in when the VI was called E g if you are currently at 2mm on the X axis and enter 5 mm for both limits then the scan w
21. a1 C28 f Ci eee Dee y s Ee A a Ey Ghosh End Function Public Function NO L As Double As Double calculates the ordinary refractive index of BBO as a function of wavelength Arguments L wavelength in microns NOP Sarao B O AL see E a F Dee E ees I ay hosp End Function Page 62 of 81 Public Function Ntheta L As Double t As Double As Double calculates the extraordinary refractive index of BBO at a given wavelength as a function of angle from the optic axis Arguments L wavelength in microns t angle from optic axis If Cos t 0 Then t pi 2 n pi Ntheta NO L Exit Function Else Ntheta NE L NO L Sqr 1 Tan t Tan t NE L NE L NO L NO L Tan t Tan t End If End Function Public Function FindThetal thetaPM As Double wp As Double wl As Double phil As Double As Double calculates the angle of the ki wavevector to kp Uses the Newton Raphson method where the derivative of the function dk the phase mismatch we want equal to zero is approximated by a central difference Arguments thetaPM the angle of the pump from the optic axis wp the pump angular frequency wl the angular frequency of kl i phil the azimuthal angle of kli around the pump beam Dim iter As Integer Dim w2 As Double theta As Double thetal As Double theta2 As Double Dim kp As Double k2 As Double kl 1 As Double kl 2 As Double kl As Double L1 A
22. ails of the coatings were not available and so could not be incorporated into the analysis 0 7 0 6 m m horizontal vertical 0 5 e w ave no AR 0 4 0 wave no AR transmittance 0 3 0 2 0 1 80 60 40 20 0 20 40 60 80 angle or incidence degrees Figure 3 5 Transmittance through BBO crystal optic axis lies at 35 to the normal in the horizontal plane The solid lines show those predicted for the transmittance with no loss or AR coating on the crystal Additional calculations assuming loss and a changing optical path through the crystal due to the rotation did not improve the fit of the theoretical curves Moreover for the e wave the actual transmittance was greater than the theoretical prediction so a straight through loss could not be obtained on that basis So in the absence of details Page 38 of 81 of the AR coatings it was assumed that the transmittance across the media boundaries is unity and that the measurements indicate only loss in the crystal This gives losses of 0 13 0 02 and 0 07 0 02 for the o and e waves respectively for the measurement at normal incidence In order to improve on these results further measurements would be needed Since we are principally concerned with the transmittance of downconverted light we might dispense with the HWP if we can ensure that incident light is vertically polarised i e corresponds to an ordinary ray Facilities exist at NPL for me
23. angles Page 25 of 81 3 Quantum efficiency measurements 3 1 Measurement of quantum efficiency Since the downconverted photons are produced in pairs the existence of one implies the existence of the other As the energies and momenta of both photons are correlated if we detect one of them at a particular point we can predict where its twin should be at that time We can exploit this fact to measure the quantum efficiencies of photodetectors illustrated schematically in Figure 3 1 N pig rig Irpef AAVV No Mig Nour Toig Tour P pump Nonlinear crystal Nour Hour four P Figure 3 1 Schematic of quantum efficiency measurements using parametric downconversion Adapted from xvi Consider each twin of the downconverted photon pairs being sent to different detectors which we will refer to as the trigger and the device under test DUT Each detector will record a certain number of the correlated photons N 7 f0 P trig ig trig and 3 1 N pur url pur Page 26 of 81 where Mrigger and Nour are the quantum efficiencies Trie and Tpyr are the total transmittances of each channel and P is the number of generated pairs Every time we detect a photon on the trigger we can ask whether we also detected a photon on the DUT Since we are restricting our attention to the subset of the total number of photon pairs that make up the trigger count the number of coincidence counts Nc that we detect in this
24. as used so that separate measurements could be taken for vertical and horizontal polarisations Since the principal plane of the BBO crystal 1 e the plane containing the laser beam direction and the optic axis was horizontal polarisation in this plane corresponds to extraordinary e waves in the crystal whilst vertical polarisation corresponds to ordinary 0 waves rotation stage HP shutter aF l trap detector iris BBO crystal iris Tisa laser 702 nm Figure 3 4 Schematic of the experimental set up for the transmittance measurements The angle of incidence was varied by turning the rotation stage on which the crystal was mounted enabling measurements over a range of angles The iris following the crystal was used to ensure that the refracted beam did not move off the detector active area It was found that there was much greater fluctuation in the detector readings with the HWP in place probably due to local heating in the plate This proved to be the greatest source of uncertainty in the measurements and is reflected by the rather large error bars shown in Figure 3 5 Page 37 of 81 Also shown Figure 3 5 are theoretical predictions for the transmittance of BBO calculated using the Fresnel equations for the o and e waves these assume no loss in the crystal However the crystal had been given AR coatings on either side 345 370 nm on the input face and 345 370 600 800 nm on the output face Unfortunately det
25. asuring reflectance which would enable us to determine the efficacy of the AR coatings empirically In previous measurements xvii the crystal was mapped into different regions and the normal transmittances found The transmittance used in practice was assumed to correspond to the central region found to be 0 9201 0 0002 Although this does not give us an angular dependence the downconverted light will be emitted at about 6 to the normal it is the more accurate value and will be used in later calculations 3 6 Results In our first set of measurements the delay unit was set at 64ns giving a total delay of 72 ns with the 9308 beginning counts after an offset of 60 ns However this arrangement gave rise to a reflection at around 121 ns after the trigger Figure 3 6 a causing a higher background count In a subsequent set Figure 3 6 b the delay was set to 96 ns giving 104 ns with a span window of 80 ns which excluded the reflection The number of triggers used was 10 000 000 for each run Figure 3 6 b showing the average of 8 runs Measurements were also taken with the filter on the DUT removed However in this case the stray light swamped the detector causing it Page 39 of 81 to register zero counts at the previous output of the laser 31 mW With the laser power turned down the stray light was still found to swamp any coincidence counts Averaged counts delay 72 ns 100000 10000 1000 2 c Ss O
26. axis gives rise to two waves polarised along a slow axis In Type II a fast pump wave decays to a fast and a slow wave These are summarised in Table 2 1 below for biaxial and uniaxial crystals Table 2 1 Types of parametric downconversion The letters e and o stand for extraordinary and ordinary respectively a ical al ial i ii In our experiments we use barium beta borate crystal BBO which is negative uniaxial and so we confine the following discussion to this category of nonlinear crystal Page 14 of 81 For Type I downconversion the daughter photons are emitted on light cones coaxial with the pump beam see Figure 2 1 Since momentum is conserved correlated pairs occur diametrically opposite each other on their respective cones Nonlinear crystal Figure 2 1 Type I downconversion for a negative uniaxial crystal The pump wave is incident from the left The squares and triangles mark typical positions of conjugate pairs of correlated photons With Type II downconversion the refractive index of the extraordinary wave depends on the angle between its direction of propagation and the optic axis Hence the light cones are no longer coaxial Figure 2 2 illustrates Type II downconversion in the general case and in the particular case of collinear downconversion where the daughter photons can be emitted in the same direction as the pump Page 15 of 81 Nonlinear crystal Nonlinear crystal
27. by the suppliers for Type II downconversion This implies that the suppliers are using the alternative crystal orientation to that used in deriving 2 31 That is the effective nonlinearity is given by dig hi cos 0 sin3 Type II 4 5 using the suppliers convention for see section 2 4 The argon ion laser can be tuned to several different wavelengths so different options for downconversion are available Table 4 1 lists the results for Type II collinear downconversion using three different sources for the Sellmeier indices Photox is the company that supplied our first crystal The values shown are for normal incidence A potential problem is the discrepancy in the results up to 5 nm for the e wave and 60 nm for the o wave Figure 4 5 shows the calculations for the lab angles of the emergent downconverted light cones with a pump at 514 5 nm using Ghosh as in Appendix A The first figure shows the collinear case the smaller cone corresponds to the e wave whilst the second shows what happens when the e wave is changed in Page 50 of 81 wavelength by 5 nm It may therefore be necessary to determine the wavelengths of the downconverted light in the collinear case for our crystal experimentally Table 4 1 Wavelength options for Type II collinear downconversion normal incidence Ghosh Kwiat Photox Alternatively we can choose a wavelength and try to find the phase matching angle to achieve
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29. dence by N Walker NPL Note the shift towards the left of the transmittance peak for the trigger filter as the angle of incidence is increased Page 43 of 81 The largest relative uncertainty in photon counts is due to the coincidence and accidental counts This can be improved by increasing the number of triggers counted as noted before in section 3 2 1 3 6 2 Correcting for the transmission of the DUT channel In section 3 2 2 the validity of approximating the transmittance from the centre of the crystal to the square root of the total transmittance was discussed Calculations xvii based on this approximation give the transmittance of the downconverted light as 0 983 0 005 This incorporates a factor accounting for the angle of the emitted light The fractional uncertainty in this figure is 0 5 as compared with the average fractional uncertainty of 0 08 found in the photon number measurements above Clearly the transmittance through the crystal contributes the greatest uncertainty at this point Since in practice a detector will be used in conjunction with a filter and lens we can treat all the optical elements in the DUT path as a single system and find the QE for this Using the value for the transmittance given above together with the average of the results found in the last section 1 e 0 191 for the total QE we have 0 194 0 001 which corresponds to a fractional uncertainty of 0 5 Taking account of the filters a QE of
30. detection efficiency PDE of a photon detection module In practice this may also include any filters or focussing lenses used In particular the manufacturers of the APDs used in our experiments specify QE for the APD head and PDE including the electronics Two detectors A and B say are used in this arrangement and aligned to intercept the downconverted pairs so that if a photon arrives at one detector its twin should arrive at the other Every time there is a count on one detector say A it can be asked if there is a corresponding count on B The number of coincidences will then be the quantum efficiency of B times the number of counts on A This has the advantage over conventional methods in that a calibrated source is not needed the measurement is absolute and can be performed with uncalibrated detectors A second application is the measurement of the spectral radiance of a light source demonstrated by Migdall et al vi In the previous case the downconverted photons were produced spontaneously However PDC can be stimulated by injecting a source of photons with the same energy and momentum as one of the downconverted photons the signal say In this arrangement the system is a parametric amplifier with energy from the pump being converted into the signal and idler fields The radiance of the source can then be measured from the increase in the number of idler photons The attractive feature of this method emerges when the stimu
31. e is a dependence on the pump intensity Any fluctuations in the pump between runs would therefore introduce uncertainties in the downconverted number densities In practice this can be overcome using a stabilised laser and making these measurements close together so that alignment conditions do not change Also since the number densities in 4 2 are averages we should take measurements over fairly long counts so that the fractional uncertainties are reduced This should also help to average out any instability in the pump 4 4 2 Detector linearity Since 4 2 involves a ratio of the idler number densities the absolute values are not significant This implies that the detector used does not actually need to be calibrated the only requirement is that it gives a linear response over the range of photon counts In practice this may be an issue since photon detectors and the counting electronics have a dead time during which they cannot register a second photon At high photon arrival rates counts may be lost so the linearity of the photon counting modules must be characterised 4 4 3 Transfer function of optics In order to image the radiant source and collimate it onto the nonlinear crystal we need some kind of imaging optics In section 4 2 we proposed using an off axis parabolic mirror In order to test the method we envisage using a black body source since this is a standard reference For the technique to be generally applicable to
32. e physicists notably Einstein who argued that quantum mechanics must be an incomplete description of nature as such action at a distance would contradict Special Relativity According to quantum mechanics knowledge of such dynamical variables is usually limited by the Heisenburg Uncertainty Principle Thus measuring the state of one entity means instantaneously determining the state of its separated but correlated partner which would seem to imply a faster than light signal In fact since no actual information 1s communicated over spacetime in this way Relativity theory remains intact and the phenomenon of entanglement has been verified experimentally 1 Quantum metrology seeks to exploit the use of correlated photons to develop measurement techniques in the photon counting regime that do not rely on existing calibrated standards and which can be tied to fundamental constants Current radiation scales are based on cryogenic radiometry 11 This is a technique based on electrical substitution radiometry ESR in which the heating effect of an optical source is compared to electrical heating Thus the primary scales are based on the SI unit of the Page 5 of 81 Ampere Cryogenic radiometers are then used to calibrate trap detectors which in turn are used to calibrate filter radiometers The latter are used to measure spectral radiance By cooling to temperatures below 20 K cryogenic radiometry has enabled the realisation of detector scales w
33. ead Time of Single Photon Counting Photodiodes A Comparison Between Two Measuring Techniques Metrologia 37 625 628 2000 Migdall A L Castelletto S Degiovanni I P Rastello M L Intercomparison of a correlated photon based method to measure detector quantum efficiency Appl Opt 41 15 2914 2922 2002 Page 79 of 81 1x x1 xii xiii xiv xv xvi xvii xviii xix Dauler E Migdall A L Boeuf N Datla R U Muller A Sergienko A V Measuring absolute infrared spectral radiance with correlated photons New arrangements for improved uncertainty and extended IR range Metrologia 35 295 300 1998 Butcher P N Cotter D The Elements of Nonlinear Optics Cambridge University Press 1990 Louisell W H Yariv A Siegman A G Quantum Fluctuations and Noise in Parametric Processes Phys Rev 124 1646 1654 1961 Kwiat P G Nonclassical Effects from Spontaneous Parametric Down Conversion Adventures in Quantum Wonderland PhD thesis University of California at Berkley 1993 Kleinman D A Phys Rev 126 1977 9 1962 Dmitriev V G Gurzadyan G G Nikogosyan D N Handbook of Nonlinear Optical Crystals Springer Verlag 2nd ed 1997 Eimer et al J Appl Phys 62 1968 1987 Migdall A L Correlated photon metrology without absolute standards Physics Today 52 1 41 46 1999 Cheung J Y Vaughan M P Mountford J R Chunnilal
34. ens the window shown in Figure C 3 If the Connected light does not come on check that the USB cable from the 9308 is connected to the PC Page 69 of 81 Coincidence counter Connected Counting OffsetSec 6 0000E 8 Span5er 8 0000E 6 Number of triggers 10000000 Triggers counted 10000000 Events counted integral 1969206 User Lik Enable stages Ka Y MOVE mms or degs by clicking on Abort counts 1500 0 1600 0 1400 0 1200 0 1000 0 agg 0 600 0 400 0 200 0 l l l l l l l l I l l l l l l l 60 65 F 75 o0 85 90 95 100 105 110 115 120 125 130 135 140 Time ns error out Moving Rotation Tilt o oo 0 00 0 00 0 00 Figure C 3 Coincidence counter 9308 window The offset span and number of triggers are set in the top left hand boxes Whilst the counter is running the Triggers counted box and the graph are continually updated At the end of the scan the Events counted box is updated This is the total number of events from the DUT counted during the scan If necessary the scan can be stopped Save data saves the parameters user date and data from the histogram in a single column Note that since the histogram has 65536 bins the saved file will have too many rows to be opened directly in Excel If this is required the file must be opened in Notepad and the parameters removed Page 70 of 81 If necessary
35. et al vi at NIST by taking the integral of the product of the source and pump profiles and normalising this to the pump In the NIST experiments this was complicated by the fact that they were using non collinear downconversion It is hoped that in our proposed experimental set up using collinear downconversion and encompassing the pump entirely by Page 54 of 81 collimated source radiation will simplify the analysis How accurately this can be done will depend on the profile we can get for the collimated source The second aspect should be fulfilled automatically for a particular pair of wavelengths in the collinear case The question will be how accurately we can know what those wavelengths are This will involve how the crystal is cut how it is rotated and what filters are used Also predicting these wavelengths can be difficult due to the discrepancy in sources for the Sellmeier indices for BBO discussed in section 4 3 Again the collinear arrangement with the collimated source should simplify the analysis for the angular overlap The uncertainties here will arise in how well we can align the optics to achieve this Having considered how well the source and pump overlap spatially angularly and spectrally we can consider where the mean point of downconversion will be necessary for establishing the transmittance of the crystal For our arrangement this should again be the centre of the crystal The problem that emerges 1s then c
36. ght cone through a lens enabling concentric red and green rings to be seen see Figure 2 5 The PDC could be optimised by eye by adjusting the HWP In the second set of experiments new lenses were used and the distance to the crystal was increased in order for the PDC to be focussed properly The distances measured to the filters from the crystal were 119 cm and 146 cm for the DUT and trigger respectively 3 4 2 Alignment using automated stages With the BNC cables connected up to the 994 counter the counts on the trigger detector can be maximised using a LabView routine that moves a given axis around an origin point and plots a graph of the counts The most effective procedure was found to be as follows First the rotation axis is scanned about a degree or so either side of its original position in increments of 0 1 usually a degree is sufficient if the alignment by eye was good The stage is then moved to the angle of maximum count Next the horizontal x and vertical y axes are scanned at increments of 0 1mm the limits being typically a few mm either side of the start point Again after each scan the stages are moved to the maxima The process is then repeated iteratively with Page 35 of 81 smaller limits and finer increments until no further improvement can be made Figure 3 3 shows some typical scans At this point it is useful to check that the half wave plate is optimised for maximum downconversion This is done by ta
37. glects the uniformity and stability of the crystal which may vary by a greater extent than this It may need to be established whether the local heating of the crystal causes transient changes in the transmission or permanent optical damage The temperature dependence of the refractive indices also needs to be quantified Given any spatial changes in uniformity another issue would be wandering of the pump beam so this should be quantified A further complication arises through the use of AR coatings Generally the crystal will have an AR coating at the downconverted wavelengths on the output face and a Page 56 of 81 coating at the pump wavelength on the input side This lack of symmetry limits how accurately the transmittance from the centre of the crystal can be modelled by the square root of the total transmittance With detailed information about the thin films used 1 e number thickness and refractive indices of the layers it may be possible to derive more accurate theoretical predictions using Fresnel s equations Alternatively the reflectivities at the wavelengths of interest might be determined empirically In all cases a robust experimental set up 1s necessary to establish the required uncertainties in the losses and ensure reproducibility Quantifying the uncertainties associated with the alignment is difficult in practice The proof of the alignment and alignment procedures lies in the reproducibility of the measurements I
38. hannel Tpuyr being corrected for The measurements took place four days apart and some re aligning had to be performed on the second set In the event this proved only to be a slight adjustment on the rotation stages In both cases the laser output was light stabilised at 31 mW The fractional uncertainties in the QE calculations are 0 07 and 0 09 as calculated using 3 5 We note a discrepancy of 0 002 between the measurements This may be due to such changes in the alignment that different parts of the detector active areas were being irradiated since these are not in general uniform Also the slight change in rotation angle may mean that the PDC is incident at a different angle on the filters in front of the detectors Page 41 of 81 Table 3 2 Values and uncertainties for photon counts earlier set Uncertainty Value absolute 1886457 1593 _ 27256 Eaa 0 190 0 0016 Table 3 3 Values and uncertainties for photon counts later set Uncertainty Value absolute Nc 1895768 Nace 16874 N pig 10000000 Nyaise 206852 561 0 192 0 0013 The angle of incidence on a filter 1s important for three reasons Firstly refraction through the filter will displace the transmitted light leading to the PDC being imaged on a different region of the detector active area Secondly at different angles the spectral transmittance of the filters is changed see Figure 3 7 which may mean that we are be registering p
39. he number of stray photons both detectors are enclosed in light tight boxes and filters are used to select out the required wavelengths However in the initial experiments performed the boxes were not used during alignment as the original designs exceeded the maximum allowed torque on the tilt stages due to the arrangement of the lens mounts and detectors on the base plates Subsequently the boxes were redesigned to achieve a more manageable arrangement and the light tight boxes were used Photographs of the actual set up can be seen in Appendix D Page 33 of 81 Two EG amp G SPCM ARQ series APDs are used as detectors Appendix E The output from the trigger is connected to the start input of an EG amp G Ortec 9308 Picosecond Time Analyzer This initiates a temporal window or span for counting subsequent events read on the stop input These arrive from the DUT after the signal has been fed through two EG amp G Ortec 425A delay units The units can produce delays of up to 64ns to compensate for the dead time of the 9308 although when the devices were tested an extra 8 ns delay was found to be added regardless of the setting This was probably due to the delay in the additional BNC cables required to connect the units An offset can be set on the 9308 window before subsequent stop inputs start to be counted to allow for the delay Any coincidence counts should be recorded as a narrow peak occurring at a fixed time after
40. hotons at a different wavelength If the effective transmittance curve of the trigger filter 1s shifted in one direction then we need to ensure that the Intermediate result without correcting for the transmittance Page 42 of 81 transmittance of the DUT filter covers the conjugate wavelength which will be shifted in the opposite direction Lastly the QE is a function of wavelength so as a consequence of the change in spectral transmittance the measured spectral profile will be correspondingly distorted To check the angles of incidence a HeNe laser is set up and reflected through the crystal on to the filters and the displacement of the reflections measured It was found after the later measurements that there was an angle of incidence of 3 2 on the trigger filter and 2 4 on the DUT In future experiments this problem should try to be rectified This may involve rotating the crystal or using a new crystal cut differently since as illustrated in Figure 2 5 and Figure 2 6 as the pump makes a smaller angle with the optic axis the chromatic dispersion increases making it easier to resolve a particular wavelength DUT black amp Trigger grey filter transmittance 60 00 50 00 40 00 30 00 Transmittance 20 00 10 00 685 00 690 00 695 00 700 00 705 00 710 00 715 00 wavelength nm Figure 3 7 Transmittance measurements of the filters used in front of the DUT and trigger at different angles of inci
41. idler photons to be in the visible range where detection technology 1s better Source idler Pump Nonlinear crystal 2 signal Figure 4 1 Schematic of parametric amplification due to stimulated parametric downconversion in a nonlinear crystal Adapted from xvi In practice 4 2 needs to be modified to take account of transmission losses the transfer function of the imaging optics and how well the source is overlapped with the pump in the crystal Subsuming all these factors into a single term T 4 2 becomes 1 NOx l _ No da T N Oor These factors are discussed in section 4 4 Page 46 of 81 4 2 Experimental set up The proposed scheme will employ Type I collinear downconversion using a copper point blackbody CuBB as the source to be measured A schematic of the experimental set up is shown in Figure 4 2 The source radiation 1s to be collimated and will enter the BBO crystal collinearly with the pump beam A filter selects out the visible photons correlated with the stimulated radiation and an aperture ensures that it is only the collinear downconverted light the detector receives A shutter is used to switch off the source so that the spontaneous downconversion can be measured The half wave plates are used to achieve the correct polarisation for the PDC Ar laser HWP beam splitter BBO crystal filter aperture lens detector __ shutter HF polariser collimating optics
42. ields arises due to the nonlinear polarisation of the medium As shall be seen whilst the classical analysis predicts the growth of the signal and idler fields it does not account for the spontaneous downconversion that seeds the process in the first place For this a quantum mechanical account is needed Page 8 of 81 First a classical analysis is followed in order to gain some insight into how PDC depends on the nonlinear polarisation The argument follows along similar lines to the analysis of second harmonic generation given in x 2 1 PDC as three wave mixing We start by assuming the frequency components of the field are plane waves of the form Eo o sr exp r_ 2 1 where E a r is the electric field A r 1s the amplitude and r is the position vector Adopting the slowly varying envelope approximation we can write the nonlinear wave equation in the frequency domain as k V w ae lo 7 exp Cik r 2 2 where 4y 18 the permeability of free space and P a r is the nonlinear polarisation In the case of PDC we are only concerned with the second order nonlinear polarisation Pp r given by x PP w r 7200 0 Eo rE or 2 3 Page 9 of 81 where amp 1s the permittivity of free space and Ya 2 is the second order susceptibility tensor The argument of yP indicates that is the resultant of the components at and So using 2 1 for the signal frequency we have POW r Ey oO O o
43. ight hand edge is used for alignment Figure D 5 The detector stages seen from the front The DUT is on the right and the trigger on the left The filter has been removed from the DUT in the last round of measurements The beam stop in the middle terminates the UV pump beam Page 74 of 81 Figure D 6 Close up of the BBO crystal used in the experiments The crystal is mounted on rotation and tilt stages to enable adjustment for the normal incidence of the pump beam Page 75 of 81 SPCM AQR series APDs Specifications for SPCM AQ series Data taken from www perkinelmer com opto PerkinElmer power cable total resistance Case operating temperature Photon detection efficiency Pd 400 nm 650 nm 830 nm 1060 nm 400 nm 650 nm 830 nm 1060 nm Pd variation at constant casetemperature 2h 25 C Pd variation 5 C to 40 C case temperature Dark Count E ia dark count variation at constant case temperature 6 hrs 25 C for 4 5 6 Page 76 of 81 Average dark count variation at 5 C to 40 C case temperature Dead Time Count rates below 50 i ns 5Mc s Output count rate before Mc s saturation Bien a i S a S o e e Cee a Pena A SO A A Settling time following power up 1 stability 1 meg counts sec and 25 C Threshold setting required on counter for digital output pulse terminate in 50 Ohms i ee Width Disable TTL Low Enable TTL High Gate Threshold Voltage V
44. ill be from 3 mm to 7 mm Note 1f scanning a rotation axis these units become degrees Page 68 of 81 The Counts step box specifies how many counts which are then averaged will be performed each time the stage stops A high count reduces the noise but increases the time for a scan to be performed The Number of triggers and OffsetSec boxes are only used in conjunction with a 9308 scan For a relatively fast scan with reasonable noise 50000 triggers 1s advised The OffsetSec is the time the 9308 waits after being triggered before it starts counting The counting span is set at 80 ns and is not configurable from this window To start the scan click on Scan If the scan needs to be interrupted click on STOP Note that it may be a few seconds before this takes effect Figure C 2 shows a typical scan Save data saves this as a spreadsheet file with the name of the axis with the position and counts in two columns Whilst the window is open further scans are always relative to the initial start position However after completing a scan the stage 1s not returned to its start point Clicking on Exit returns you to the move TP stages window and the new position is updated in the relevant text box Counting coincidences To perform coincidence counts with the 9308 once the detectors are aligned return to the main window and select COINCIDENCES from the Mode menu This op
45. in x As Double As Double calculates the inverse sine Arguments 7 x sine arcsin Atn x Oor I x x End Function Page 65 of 81 C LabView Software User Manual LabView is a graphical programming language used to read and control equipment using a PC via some communications protocol typically RS232 C serial port or GPIB Modules written in Lab View are called virtual instruments or VIs The structure of a program becomes hierarchical as sub VIs are incorporated into other VIs Sections of code are often described as wiring diagrams In our experiments our requirements were the control of the stages to position the detectors and reading the counter electronics Ultimately the VIs will be compiled into an executable for the correlated photons project Due to its graphical nature and much of the code being hidden in multiple case structures sequence structures and sub VIs it is not practical to print out the source code Instead this appendix 1s intended as a user manual to the stage control and counter reading software Moving the TP stages and reading the 994 counter After starting the application select move TP stages from the mode drop down menu This opens the TP stage control window Figure C 1 Select the stage to be moved from the drop down box Note that_X is the horizontal axis and Y 1s the vertical Enter the distance to be moved in the MOVE box and either pres
46. it 800 nm would be a convenient choice attainable with a pump at 351 1 nm Table 4 2 since a filter radiometer see 11 is available at this wavelength for monitoring the blackbody This would allow direct comparisons of the correlated photons method to be made to a conventional technique Achieving these angles with the current crystal would necessitate rotating it Page 51 of 81 Table 4 2 Calculation of phase matching angle to achieve 800nm downconversion using a pump at 351 1 nm e wave um 0 wave um theta degs Ghosh 0 6257 0 8000 52 99 0 8000 0 6257 45 30 0 6257 0 8000 52 86 Photox 0 6257 0 8000 53 08 0 8000 45 46 Extraordinary j Ordinary x A T a be A b s 10 Figure 4 5 Laboratory angles relative to the pump beam for Type II downconversion with a pump of 514 5 nm a e wave 665 4 nm b e wave 670 nm The dots indicate the light cones seen on exit The smaller cones correspond to the e waves From this direction the optic axis would be projected along the vertical line which would therefore be rotated by the azimuthal angle in the lab 4 4 Sources of uncertainty 4 4 1 Pump stability A first point to note is that in deriving 4 2 from 4 1 we assumed that the sinh xt term would cancel for the number densities with the signal on and off However Page 52 of 81 comparing x to the analogous gain term g in 2 10 for the classical analysis we see that ther
47. ith uncertainties of 0 005 The uncertainties in the scales for radiant sources are currently 0 5 The quantum methods use correlated pairs of photons produced by parametric downconversion PDC in a nonlinear crystal In this process an incident pump photon decays into two daughter photons conventionally called the signal and the idler with energy and conservation being conserved This can be expressed as Oo 0 0 a 1 1 where the are the angular frequencies and the k are the wavevectors of the pump and daughter photons These equations are known as the phase matching conditions as they also arise out of a classical analysis of three wave mixing section 2 One of the ways in which this can be used is to determine the quantum efficiency of a photo detector first demonstrated in 1977 111 and corroborated by other groups since then iv v A brief discussion about terminology would be useful here Quantum efficiency usually refers to the efficiency with which a photodiode converts incident photons to electrons In the case of say an avalanche photo detector APD the situation is complicated by the existence of gain in the device Added to this there is the transfer function of the electronics of the detector and any counting units In this Page 6 of 81 dissertation and in much of the literature concerning the techniques described here QE is considered in a more general way which could be described as the photon
48. king readings with the HWP at different positions and fitting the data to a cosine squared curve 994 scan over DUT rotation stage increment 0 1 994 scan over DUT horizontal stage increment 0 1 mm 1 15 2 25 6 1 distance from origin degrees distance from origin mm Figure 3 3 Alignment scans of the DUT axes On the left is a rotational scan taken at increments of 0 1 On the right is a scan over the horizontal axis with an increment of 0 1 mm These increments are used on a first scan of an axis With the detectors still connected to the 994 counter the DUT can be aligned to ensure that it is also seeing downconverted light Only the first iteration of the above process need be carried out on the 994 since the fine tuning is achieved by connecting to the 9308 and maximising the coincidence counts using a similar procedure 3 5 BBO transmittance measurements oblique incidence The transmittance of the BBO crystal was measured using the set up shown in Figure 3 4 The T1 Sapphire laser was mode locked at 702 nm with a pulse width on the order of picoseconds The use of mode locking reduces interference effects from the two faces of the crystal due to the short coherence length of the laser pulses compared to CW xix Since the detectors have been shown not to temporally resolve the Page 36 of 81 pulses the output looks like CW The output light from the laser was vertically polarised A half wave plate w
49. l C J Correlated Photon Metrology of Detectors and Sources Proc SPIE submitted 2003 Anderson V E Fox N P Nettleton D H Highly Stable Monochromatic And Tunable Optical Radiation Source And Its Application To High Accuray Spectrophotometry Appl Opt 31 536 545 1992 Hartree W S Theocharous E Fox N P A Wavelength Tunable Quasi CW Laser Source for High Accuracy Spectrometric Measurement in the 200 nm to 500 nm Region Proc SPIE 4826 104 112 2003 Page 80 of 81 20 G Ghosh J Appl Phys 78 6752 1995 Page 81 of 81
50. larisation An iris 1s used to narrow the beam and a polariser to clean out any unwanted vertical components The downconversion could then be turned on and off by the rotating the polarisation with a half wave plate The BBO crystal is cut with the optic axis at 35 to the normal and aligned so that the optic axis 1s in the horizontal plane This gives Type I downconversion for the degenerate case of two ordinarily polarised photons at 702 nm when the pump beam is polarised horizontally After solving equations 2 19 and 2 20 using the Sellmeier equations for BBO given in Appendix A both downconverted photons are predicted to emerge on a light cone coaxial with the pump beam with a half angle of 6 1 Page 32 of 81 Trigger TIL Argon ion laser Raine l BBO ar 51 10m ae seas rd i crystal a i L i Filters and lenses 425A 308 delay units time analyzer Figure 3 2 Schematic of the experimental set up for quantum efficiency measurements The detector and lens assemblies are set on plates mounted on x y rotation and tilt stages to allow for alignment to the downconverted beams The stages are controlled remotely via computer using software written in LabView Appendix C Additionally the detectors themselves are mounted on further x y and z stages fixed to the plates In the case of the DUT these are also automated to allow for spatial mapping and fine tuning the focussing In order to reduce t
51. lated output is compared to the spontaneous output with the source shuttered The spontaneous Page 7 of 81 downconversion can be considered to be stimulated by the background vacuum fluctuations of one photon per mode This means that the spectral radiance of the source can be expressed in terms of photons per mode which in turn is expressible in terms of fundamental constants Essentially the vacuum fluctuations are being used as a measurement standard a standard that is clearly universal One technical advantage of this is that if the source is in the infrared it can be arranged to have the idler in the visible where the detection technology is more advanced Current work on QE and spectral radiance measurements has realised uncertainties of the order of 0 5 v vii viti and 1 v1 1x respectively Ongoing research is investigating how these uncertainties may be reduced in order to make the use of correlated photons a viable technique After reviewing the theory of parametric downconversion in section 2 measurements of QE conducted at NPL are reported in section 3 in which an uncertainty of 0 5 was achieved In section 4 a proposed plan for spectral radiance measurement is described and the possible sources of uncertainty discussed 2 Parametric down conversion The phase matching conditions 1 1 can be derived through a classical analysis in which PDC is viewed as a three wave mixing process where coupling between the f
52. nal and idler fields by the subscripts s and i as before the time dependent solutions for a and a can be written according to vi as a q xpe ov h coshx exp bai sinh 2 15 al C plo F coshx exp 4i sinhx i where xis a gain coefficient and is a phase term introduced by the pump Calculating the expectation values of the photon number operators given in 2 13 we have N O Vo cosh x N sinh K 2 16 N O Vo cosh x N sinh K at Comparing this result to 2 11 derived from the classical analysis we see that apart from now being in the time domain 2 16 predicts the growth of the signal and idler Page 13 of 81 fields even if the initial number densities are zero due to the unity terms These terms are a direct result of the commutation relations 2 14 and correspond to spontaneous downconversion Since the dimensions of 2 16 are photons per mode we interpret the spontaneous downconversion as in fact being stimulated by one photon per mode due to vacuum fluctuations In section 4 1 we shall see how this result enables us to make spectral radiance measurements in terms of fundamental constants 2 3 Phase matching In order to meet the phase matching conditions 1 1 we need to exploit the birefringent properties of the nonlinear medium In general PDC is usually only possible in two cases which we call Type I and Type II With Type I downconversion a wave polarised along a fast
53. on arrival times are assumed to be Poisson distributed then the uncertainties can be taken as the standard deviation 1 e the square root of the mean In this case the relative uncertainties should decrease as one over the square root of the number of counts However it was found that the measured standard deviations differed from the square roots of the averaged photon counts It was suggested that this may be due variation in the pump intensity Bearing in mind that the laser is light stabilised to give a constant optical output power it is possible that fluctuations are arising in intermediate optical elements as was found in the transmittance measurements reported in section 3 5 where the introduction of the HWP introduced significant noise into the measurements Even if this 1s the case the effect of taking counts for longer should average these fluctuations out and reduce the relative uncertainties In principle it should be possible to choose a number of counts to take that will make these uncertainties arbitrarily small The actual standard deviations can be found empirically thereby establishing what number of counts will be necessary The counting electronics is another area that requires investigation The problems with reflections due to impedance mismatching were discussed in section 3 3 The Page 58 of 81 reflections themselves did not cause particular problems since they were not overlapping with the coincidence peak and c
54. onsidering reflections at the boundaries Rotating the crystal will change the spectral transmittance of any AR coating on the exit face Also we may wish to use the same crystal at different frequencies This means that approximations used in section 3 6 2 may no longer be valid and further analysis may be required Page 55 of 81 5 Discussion and conclusions The current research in quantum metrology using correlated photons offers a new approach to the determination of radiometric scales in the photon counting regime Whereas these scales are currently based on calibrated detectors via cryogenic radiometry and known sources such as blackbodies or synchrotron radiation the use of correlated photons offers a technique to determine detector and source scales directly and absolutely If this technique is to compete with cryogenic radiometry as a method of realising primary radiometric scales then improvements in accuracy of one or two orders of magnitude are necessary Improvements by a factor of ten or less may still make this a viable technique for measurements in the photon counting regime For the quantum efficiency measurements the largest current uncertainty is associated with the transmission through the crystal The data from Table 3 1 for transmittance absorption and scattering suggests that realisable uncertainties for the loss in the crystal could be as low as 0 02 as compared to 0 5 used in section 3 6 2 However this ne
55. ould be safely excluded from the integration of the counts An interesting feature that as yet has not been explained is the increase in the background level after the first reflection see Figure 3 6 a Supposedly if all the counts were being reflected there should be the same level of noise before the reflection as there is after It is not obvious that this will affect the accuracy of the measurements but this should be investigated Given that the 9308 counter 1s designed to take NIM pulses as input and that the pulses used were in fact attenuated and inverted TTL pulses it would not be surprising if the performance of the electronics was compromised It is hoped that the arrival of the bespoke TTL NIM converter will improve the reliability of the counting system Many of the points discussed above will also be applicable to the spectral radiance measurements once the experiment is set up Particular details relevant to this method have already been discussed in section 4 4 As with the QE measurements good alignment and a robust experimental set up will also be crucial In addition to the alignment of the laser optics and maximising counts on the detector where a similar procedure to that discussed previously may be used there is also the alignment of the off axis parabolic mirror to be considered Possibly this might also be mounted on automated stages An alignment procedure may then be to optimise the detector for spontaneous PDC and
56. p beam Dim w2 As Double Dim k2 As Double kl As Double Dim alpha As Double w2 wp wl alpha arccos Cos thetaPM Cos thetal Sin thetaPM Sin thetal Cos phil kl Ntheta 2 pi C wl alpha wl C Poe NOL Fads Fae We ee FindTheta2 arcsin kl Sin thetal K2 End Function Public Function FindLabl thetaPM As Double theta As Double phil As Double wl As Double As Double calculates the laboratory angle via Snell s law for the ki vector Arguments i thetaPM the angle of the pump from the optic axis i theta the angle of kli to the pump beam inside the crystal r phil the azimuthal angle of ki around the pump beam i wl the angular frequency of kl Dim alpha As Double n1 As Double alpha arccos Cos thetaPM Cos theta Sin thetaPM Sin theta Cos phil nl Ntheta 2 pi C wl alpha FindLabl arcsin nl Sin theta End Function Page 64 of 81 Public Function FindLab2 theta As Double w2 As Double calculates the laboratory angle via Snell s law for the k vector Arguments theta the angle of k2 to the pump beam inside the crystal w2 the angular frequency of k2 Dim n2 As Double n2 NO 2Z pi C 7 w2 FindLab2 arcsin n2 Sin theta End Function Public Function arccos x As Double As Double calculates the inverse cosine Arguments Cosine arccos Atn Sgqr 1l x x x End Function Public Function arcs
57. s Enter or click on the MOVE button Depending on whether the stage is translation or rotation the units will be either mm or degrees The position of the stage 1s then updated in the corresponding text box below If the stages need to be stopped immediately click on KILL Page 66 of 81 CONTROL TIME AND PRECISION STAGES Which Stage TRIGGER xX STAGE TE MONE SCAN KILL EXIT Movi MOVE trims or degs o DUT COUNTS Plot A error out 10000 0 Trigger Y Trigger Rotation 0 00 0 00 ee Trigger Trigger Tilt a 6000 0 o 00 0 00 a 3 4000 0 DUT DUT Rotation 0 00 0 00 S BUT Y BUT Tilt 00 i maxDLIT Porro penta a 100 a 0 00 0 00 Soar 10000 TRIGGER COUNTS Foto maxTrigger 994 COUNTER 10000 0 ooo enable counters l i Running Poua pulse specs Cc P g i k 6000 0 delay secs DLIT counts sec a s 0 500000 acini nan 3 4000 0 E pulse width secs Saso Trigger counts sec 2000 0 x 0 0 i 0 100 994 read Figure C 1 Window for controlling the TP stages and reading the 994 counter If a 994 count is required at the end of the move the enable counters button must be toggled to on when it lights up The individual counts are returned in the boxes below this whilst a record of the counts for each detector is plotted on the charts To take a count without moving just enter zero in the Move box Clicking on Save Data saves this data in
58. s Double Dim alphal As Double alpha2 As Double alpha As Double Dim dkl As Double dk2 As Double dk As Double w2 wp wl theta pi 2 try to find tve solution for theta dt 0 00001 kp Ntheta 2 pi C wp thetaPM wp C k2 NO pi Cf We w2 E Ll 2 pi C wl wavelength of ki For iter 1 To 20 theta2 theta dt thetal theta dt alpha arccos Cos thetarM Cos theta Sin thetaPM Sin theta Cos phil alphal COSC Oil alpha2 arccos Cos thetaPM Cos theta2 Sin thetaPM Sin theta2 COs pha 1 arccos Cos thetaPM Cos thetal Sin thetaPM Sin thetal Page 63 of 81 kl Ntheta Ll1 alpha wl C ki 1 Ntheta L1 alphal wl C ki 2 Ntheta il alone wi foe dk o Jee 5 K2 A ke ea os ole ee COS thecal kel A ded dkl kp kp k2 k2 2 kp kl 1 Cos thetal kl 1 kl 1 dk2 kp kp k2 k2 2 kp kl 2 Cos theta2 kl 2 kl 2 theta theta 2 dk dt dk2 dkl Next iter FindThetal theta End Function Public Function FindTheta2 thetaPM As Double wp As Double wl As Double phil As Double thetal As Double As Double calculates the angle of the k2 wavevector to kp Arguments thetaPM the angle of the pump from the optic axis wp the pump angular frequency l wl the angular frequency of ki phil the azimuthal angle of ki around the pump beam f thetal the angle of ki to the pum
59. s seen by the signal and idler respectively Converting 2 9 into intensities averaging to eliminate the phase dependent term and using the relation 7 n n We arrive at an expression for the mean number densities of the photons in the downconverted fields N V cosh V sinh classical result 2 11 N V cosh V sinh z As we shall see the quantum mechanical result will have a subtle but important difference 2 2 Quantum mechanical analysis A quantum mechanical model of parametric processes was developed in 1961 by Louisell et al xi The Hamiltonian of the modes in the nonlinear medium can be written H D E B HdV Za i 2 12 e eE E B H dy 25 2 V V P Edy so that the second term involving the second order polarisation 1s treated as a perturbation After quantizing the growth of the downconverted fields is described by creation and annihilation operators aA and a t such that at t 0 Page 12 of 81 Ato Ayo Nio elie 2 13 where N 1s the number of photons in the Ath mode and N 9 gt denotes the state with this eigenvalue The operators a and a are so named because they transform the Nzo states into Nin 1 and Nig 1 states respectively A significant aspect of the quantum mechanical treatment arises because a and a obey the commutation relations F Ca C O ia 2 14 N F a F Ca q Denoting the sig
60. stal we apply Snell s law which is a simple matter 1f the pump beam is normal to the crystal surface Figure 2 5 shows actual photographs of the light cones for BBO whilst Figure 2 6 indicates the theoretically predicted cone angles for three wavelengths Note that the central yellow spot due to the fluorescence on the UV blocking filter originates at a point after the downconversion and so would move relative to the light cones as the camera position was moved Page 19 of 81 Figure 2 5 Photographs of parametric downconversion in BBO with the optic axis at 35 to the normal and pumping at 351 1 nm The sequence of pictures shows the crystal being rotated through plus and minus 6 Thus in the top left hand picture the pump is making an angle of 41 with the optic axis in the bottom right an angle of 29 The top right hand picture shows normal incidence The bright yellow patch in the centre of each picture is due to fluorescence on the UV filter used to block the pump radiation The photographs were obtained by imaging the light cones through a lens on to a digital camera Half angles of PDC with 351 1 pump 14 12 10 w 2 D 8 650 nm gZ v 5 Onm 2 6 510 nm Oo te oO i 4 2 0 29 00 31 00 33 00 35 00 37 00 39 00 41 00 pump angle with optic axis degrees Figure 2 6 Calculations of the cone angles at different wavelengths against the angle of the pump with the optic axis Note the cross over point for
61. t was noted in section 3 6 1 that after the stages had been aligned to obtain maximum counts the filters were not normal to the crystal As mentioned in that section this can introduce uncertainties due to change in the way that the detector active areas are irradiated and shifts in spectral selectivity This may necessitate changes in the alignment procedure At present the filters are fixed to the front plates with double sided tape Since these plates may not be perfectly normal to the base plates on which the lens detector assemblies are mounted it may be to mount the filters in adjustable holders so that they can be adjusted for normality at the start of the alignment procedure After the initial round of alignment scans the filters could be checked for normality to the crystal as described in section 3 6 1 The stages could then be rotated until the filters were contra reflecting and the optimisation repeated Page 57 of 81 Also the angle of the optic axis with pump beam can be reduced either by rotating the crystal or using another cut differently At smaller angles there is higher chromatic dispersion 1 e the rate of change of cone angle with wavelength is greater This allows a given wavelength to be resolved more accurately The uncertainties associated with photon counting are much smaller than the transmittance uncertainties As improvements in transmission measurement are made these may need to be reviewed If the phot
62. th respect to the pump beam as illustrated in Figure 2 4 the momentum phase matching conditions can be written x11 k P k Ik cosa k 2 19 P Page 17 of 81 where q is the angle between the k vector and the pump Here the and 0 dependences have not been made explicit in order for 2 19 to be appropriate for both Type I and Type II downconversion It must be remembered however that k will depend on 0 the angle between the pump and the optic axis Moreover for Type II downconversion K which we take to be the extraordinary wave will depend on the angle between k and the optic axis which in turn depends on q and the angle of rotation around the pump beam 4g In this case 2 19 must be solved numerically see Appendix B Clearly is simply related to by k sina lt sina 2 20 optic axis Figure 2 4 Vector representation of phase matching Angles shown are those within the crystal The magnitude of k depends on the angle 6 For Type II downconversion taking K as the extraordinary wave the magnitude of k will depend on the angle In the pump reference frame the x axis points downwards the y axis points to the right and is the azimuthal angle around the pump Figure adapted from reference xii Page 18 of 81 The angles indicated in Figure 2 4 and used above are those inside the crystal To calculate the half angles of the light cones that emerge from the cry
63. the colours which is also seen in the photographs in Figure 2 5 Page 20 of 81 2 4 The effective nonlinearity In section 2 1 we introduced the effective nonlinearity dep In this section we derive expressions for dey for the processes of Type I and Type II downconversion in BBO We start by noting that due to the commutativity of the electric field vectors the second order susceptibility tensor can be reduced to an 18 element d tensor The form of the d tensor can usually be simplified by examining the crystal symmetries of the nonlinear medium in question If in addition Kleinman symmetry x111 holds 1 e the nonlinear medium is transparent in the spectral region of interest then further simplifications can be applied For BBO the crystal symmetry and point group is Trigonal 3m For our purposes we shall be interested in a spectral range from around 300 nm to the near infrared According to the Sellmeier equations used in Appendix A the nearest resonances of BBO are around 130 nm and 9 5 um so we are justified in applying Kleinman symmetry In this case the d tensor is x o 0 0 0 d dy d dp dp 0 ds 0 07 2 21 dis dis Ge O 0 0 where the over bars indicate negation The component of the polarisation is now given by P d Eo rEo sr 2 22 Page 21 of 81 where after making the r dependence implicit the second order electric field tensor can written out as E EE a ee E o o X 6 2
64. then we can take the square roots of the means to be the uncertainties However in our experiments the stray light is due principally to fluorescence from the UV laser and downconversion at other wavelengths Both of these sources depend on the intensity of the laser so ANacc ANc and ANyaise may be affected by the laser stability However with the laser light stabilised this should not be a major problem and assuming the uncertainties remain proportional to the square roots of the means the contributions Page 28 of 81 of these fractional uncertainties become smaller as the number of trigger counts and consequently the coincidence accidental and false counts is increased 3 2 2 Uncertainties in transmission The transmittance Tpur includes the optical losses in the crystal filters and focussing optics as well as any loss of downconverted photons on the DUT due to misalignment The uncertainty A7pu7 therefore depends on how accurately these losses can be specified 3 2 2 1 Losses in the crystal Losses in the crystal arise through reflection on the output face of the crystal absorption in the medium and scattering The first of these can be alleviated through the use of anti reflection coatings at the wavelength of the downconverted light For low reflectances the transmittance from the centre of the crystal can be approximated to the square root of the total transmittance Calculations show xvii that from the centre of
65. ulec secular ET 52 4 4 1 PUTAS COTY ceesh atest sa tetteal il ance tstaciotatat hintateatent tatontghet ae hates hundateanane 52 4 4 2 D ector ne eet Ie O AE TER 53 4 4 3 Transter TUncUon Or ODES sonnen a TOA 53 4 4 4 Overlap factor and transmission LOSSES ccccseeeeseeecccceeecaaeseeeeeeeees 54 D gt DISCUSSION GNG CONCIUSIONS cuss valent e hea tetidallay tata T Dace 56 O ACINO CUC EMCI IS rsciscca ened E E EA 60 TF PAD CTI CCS ETE OET E ean dace E O 61 As Sellmeier eguations for BBO senaisiais 61 B Visual Basic functions for phase matching calculations c ccc eeeeeeeeeeeeeees 62 Ce LabView Software User Manual issssercsiveteveevevassnieNiieyvandntisiteveneeaaiew 66 D Photographs of the experimental set up for QE measurements 664 T2 Es SPCMEAOR SENES APDS rner aN E i eee 76 D RESF T EEE T O phan aie 79 Page 4 of 81 1 Introduction In recent years there has been growing interest in the area of quantum metrology This new field has its theoretical roots in the physics of entangled or correlated states Entities e g photons may be produced in pairs or groups of higher number and their dynamic properties then remain correlated in accordance with the conservation laws of energy and momentum Measuring say the momentum of one particle means knowing instantly the momentum of the other even when they are separated over space This has historically been a point of contention for som

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